Determination of the Convective Heat Transfer Coefficient ... · PDF fileDETERMINATION OF THE...

DETERMINATION OF THE CONVECTIVE HEAT TRANSFER COEFFICIENT OF HOT AIRRISING THROUGH TERRACOTTA FLUES

Taylor A. Oetelaar1, Clifton R. Johnston21Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, Canada

2Department of Mechanical Engineering, Dalhousie University, Halifax, CanadaE-mail: [email protected]; [email protected]

Received June 2012, Accepted September 2012No. 12-CSME-69, E.I.C. Accession 3389

ABSTRACTWe experimentally studied natural convection processes inside terracotta flues as a part of a larger numer-

ical study of ancient Roman baths. The air, heated in a plenum below the wall, rose through the tubes. Twoclusters of thermocouples, equally spaced in the flues, measured temperatures throughout the thickness ofthe wall. The data from the two clusters proved to be measurably different. The resulting convective heattransfer coefficients determined using the bottom cluster, showed no dependence on the plenum tempera-ture. The measured convective heat transfer coefficient was between 6.2 and 7.6 W/m2·◦C, with an averageof 7.0 W/m2·◦C.

Keywords: convective heat transfer coefficient; natural convection; Roman baths.

DÉTERMINATION DU COEFFICIENT D’ÉCHANGE D’AIR CHAUD PAR CONVECTION ÀTRAVERS DES CONDUITS EN TERRE CUITE

RÉSUMÉNous avons étudié, à des fins expérimentales, dans le cadre d’une étude élargie sur les anciens bains

romains, le processus de convection naturelle à l’intérieur des conduits en terre cuite. L’air chauffé dansun plénum, situé au bas du mur, s’élevait à travers des tubes. Deux réseaux de thermocouples, disposésà égale distance dans les conduits, évaluaient la température à travers l’épaisseur du mur. Les données desdeux réseaux montrèrent des différences mesurables. Les résultats des coefficients d’échange d’air chaud parconvection, réalisés en utilisant le thermocouple situé en bas du mur, ne démontraient aucune dépendancesur la température du plénum. Le coefficient d’échange de chaleur par convection mesuré se situait entre 6.2et 7.6 W/m2·◦C, avec une moyenne de 7.0 W/m2·◦C.

Mots-clés : coefficient d’échange d’air chaud par convection ; convection naturelle ; bains romains.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 36, No. 4, 2012 413

NOMENCLATURE

g gravity (m/s2)h convective heat transfer coefficient (W/m2·◦C)k thermal conductivity (W/m·◦C)L characteristic length (m)Nu Nusselt numberq′′ heat flux (W/m2)Ra Rayleigh numberT temperature (◦C)∆x horizontal difference (m)

Greek symbolsα thermal diffusivity (m2/s)β thermal expansion coefficient (◦C−1)υ kinematic viscosity (m2/s)

Subscriptscon conductivee f f effectiveL characteristic lengthw wall∞ bulk

1. INTRODUCTION

The standard method for heating, ventilation, and air conditioning (HVAC) systems is using mixing air tomanage temperature, air quality, etc. More and more systems, however, are combining this with radiant floorheating and cooling. While this interpretation is relatively new, the idea of using a hot fluid beneath a raisedfloor to heat the room above was invented by the Greeks, and subsequently developed by the Romans, toheat bathing complexes (Fig. 1a). They took it one step further by including wall heating. Hot exhaust gasesfrom fires below would rise up terracotta pipes, called tubuli, inlaid in the wall. We are using computationalfluid dynamics (CFD) to investigate the largely unknown thermal environment in the caldarium, or hot bathroom, in the Baths of Caracalla in Rome [1], using the caldarium of a building built for the NOVA televisionseries [2] as a validation model (Fig. 1b). However, as the main boundary conditions of the CFD model andthe driving force behind the air circulation, an accurate assessment of the heating capabilities of these inlaidflues is necessary. This paper describes an experiment to investigate how much heat was transferred fromthe exhaust to the wall (Fig. 2). The results could have an impact on future HVAC radiant heating systems.Without fans, the Roman system was able to reach high room temperatures. With modern adaptations,this system could provide a new, more energy efficient way to heat and cool buildings. First, however, thefundamentals must be understood, namely the convective heat transfer coefficient (CHTC) of the air risingthrough the tubes.

Many of the studies on CHTCs involve a warm surface heating cool air and seek to determine howmuch heat is drawn away from the surface. Of these, few investigate scenarios comparable to the situationpresented here, including analyses of enclosures, pipes, and ducts. Most of these proposed relationshipsbetween the Rayleigh (Ra; Eq. 4) and Nusselt (Nu; Eq. 3) numbers which will be shown later to be inappro-priate for this experiment. However, it is important to note the ranges of Nusselt numbers for comparativepurposes. One of the more interesting is the study by Hatami and Bahadorinejad of a vertical flat-plate solar

414 Transactions of the Canadian Society for Mechanical Engineering, Vol. 36, No. 4, 2012

(a) (b)

Fig. 1. (a) Diagram of Roman heating system showing key features. Inset is of a terracotta flue used as a tubulussubstitute. (b) A CAD model of the replica baths built for NOVA showing the various rooms including the caldarium.

(a) (b)

Fig. 2. (a) Picture of experimental setup. (b) Picture looking down the rightmost flue showing the layers of the wall(Photos by Matt Oetelaar).


air heater [3]. Nusselt numbers ranged from 2.9 × 102 to 3.7 × 102 (1.2×1010<Ra<3.0 × 1010) for channelflow and 1.4 to 1.9 (3.0 × 104<Ra<1.2 × 105) for enclosures. This was very close to those for the Romanheating system except the surfaces are heating the air instead of the other way around. Fossa et al., useda similar setup to examine the possibility of convection to cool photovoltaic panels [4]. Nusselt numbersranged from 3.2 to 68.3 (1 × 107<Ra<1 × 1012) for one case, 13.3 to 56.6 (1 × 107<Ra<8 × 109), and13.7 to 57.4 (1 × 107<Ra<8 × 109) for the final case. Calay et al. looked at heat transfer within rectangularenclosures with different boundary conditions [5]. Nusselt numbers ranged from 7.8 at a Ra of 3.9× 108 forone configuration to a high of 74.5 at a Ra of 6.5 × 109 for another configuration. Terekhov and Terekhovmodeled heat transfer in a vertical container with various numbers and lengths of fins attached to the hotside [6]. Nusselt numbers ranged from a low of 1 with no fins and a Ra of 4 × 103 to a high of 4.5 with 20fins and a Ra of 1× 105. Rao analyses a vertical channel air heater to compare convective and radiative heattransfer [7].

Since this experiment, at its foundation, bridges two fields, it is important to look at what previous schol-ars have done in archaeology as well. Kretschmer preformed one of the most in-depth investigations ofhypocaust systems in the early 1950s. In his article [8], he chronicles temperatures in a replica system thathe built. Hüser [9] continued to perform experiments on Kretschmer’s replica focusing on the raised floorand analyzing the modern applicability of the hypocaust. Jorio [10] provides detailed information about thehypocausts of the Pompeian baths (Stabian, Forum, and Central) and then analyzes the heat losses from fourrooms in the Stabian Baths. Rook [11] utilizes thermodynamic equations to come up with an approximationof how much fuel the Welwyn Roman bath consumed. Baatz [12] investigated the effectiveness of canalheating and compared this to the hypocaust system. Basaran preformed multiple numerical analyses on thehypocaust. The first [13] used numerical heat transfer to investigate the heating system of the Small Baths atPhaselis in Turkey. The second [14] used CFD to analyze the heating system of a bath house in Metropolis.The third and final [15] is, in essence, a summary of the first two. Yegül and Couch used temperature probesin the replica bath to analyze the heat losses [2]. The issue with all of these studies, however, is no onedetermines the convective heat transfer coefficient of the tubuli which is where this study fits in.

2. EXPERIMENTAL SETUP

The basic setup for this experiment (Fig. 3a) was a section of wall placed above a plenum which housesthe heater. A wooden box faced with cement board on the inside and covered with both batt and rigidinsulation encased these. A volume of air above the plenum and next to the wall section was meant as afaux room to prevent the direct conduction between the wall and the outside. It was important to maintainmaterial consistency as much as possible between this case study and the wall we were replicating. Thetubuli used in the bath were custom made and thus unattainable. Superior Clay terracotta flue liners weresimilar in size, shape, and material to the tubuli in the replica bath. The tubuli were prismatic and were200 mm wide by 120 mm deep by 250 mm high. The flues were 216 mm wide by 114 mm deep by 607 mmhigh but the shorter edges were semi-circular instead of straight. Both the tubuli and the flues are of modernterracotta, however, the specific composition is most likely not the same. Six of the flues in an array threewide by two high with the rounded sides facing each other made up the main wall component. Lime mortarand marble tiles then covered the side facing the room substitute.

The initial heater was a Watlow 300 W finned strip heater. A thermocouple located above and behind theheater, measuring what will be referred to as the plenum temperature, was attached to a controller. The peaktemperature with the 300 W heater was 40°C which was insufficient. A larger 750 W finned strip heater wasmore appropriate as its peak temperature was 160°C.


(a) (b)

Fig. 3. (a) Cross-section diagram of the setup (Dimensions: mm; Arrows indicate air flow direction). (b) Magnifiedview of the wall cross-section showing thickness dimensions and the approximate depth of the thermocouples.

One final set of adjustments had to be made to keep the mass flow rate steady. A 31.8 mm hole was drilledin the front door of the plenum and the outside two flues were capped with rigid foam. This ensured thecorrect draw weight for the heater.

There were 16 K-type (Chromel-Alumel) thermocouples used for measurement. All were located in thecentre two flue liners to minimize the effect of the edges. They were broken down into two groups ofeight — one cluster located half way up the lower flue liner (referred to as the bottom cluster) and the otherhalf way up the upper flue liner (referred to as the top cluster). In each group, the spread was about 50.8 mmand thermocouples were located at specified depths. The eight depths were: the middle of the flue (capturingthe air temperature), on the inner surface of the flue, one-third of the flue wall thickness, two-thirds of theflue wall thickness, on the outer flue surface, half way through the mortar, on the inner surface of the marble,and finally, half way through the marble (refer to Fig. 3b).

The thermocouple placement was difficult because, unlike in many experiments, they had to be insertedduring construction. This posed two major problems. The first was ensuring that the thermocouples didnot move without drastically affecting the thermal properties. A combination of silicone and tape helped,however, some of the thermocouples may have moved slightly even with this procedure. The second wasensuring that no two thermocouples were directly on top of one another while still ensuring a tight clus-tering. Since all the materials were opaque and each layer erases the landmarks, an approximate circularpattern was employed.


An unexpected problem arose with the thermocouples on the inner surface of the flue next to the rising air.Even though the thermocouples were encased in concrete when they were mounted, the readings indicatedthat they were measuring the air temperature because they mirrored the fluctuations in the free stream read-ings. To test this hypothesis, a FLIR infrared camera took a sample of temperature readings. The infraredmeasurements proved to be drastically different from the bare wire thermocouple readings and closer to theexpected data, so a K-type infrared thermocouple mounted on a rod in the middle of the flue replaced thesurface thermocouple.

The data acquisition setup was uncomplicated. After the thermocouples measured the signals, they passedthrough an AD595 amplifier and captured by a NI PCI-6024E DAQ. The long time frame of this experiment,however, made continuous data sampling challenging. A custom LabVIEW virtual instrument (VI) programsampled 30 seconds worth of data at 200 Hz every 5 minutes. The VI then averaged the data and plottedthese points on a continuous graph.

2.1. CalibrationThe thermocouple calibration had to be completed in situ. The lower point was ascertained by allowing

the system to reach room temperature. To obtain a higher point, insulation was placed over the marble andthe inlet and outlet were both sealed with insulation. A fan was installed on the top which draws air from oneflue and blows it back down another. The result was that the temperature equalized throughout the thicknessof the wall. This provided the second point which allowed for the calculation of the calibration slopes andoffsets.

3. RESULTS

Once the methodology and equipment were satisfactory, the experiment was run on five plenum tem-peratures (60°C, 70°C, 80°C, 90°C and 100°C) for each thermocouple cluster. Three temperatures werealso duplicated — two for the bottom cluster and one for the top — to determine repeatability. The reasonwe chose the plenum temperature is that it is the only controllable variable. Most experiments measuringCHTCs use the wall temperature or heat flux as their reference but neither are applicable here. The walltemperature is constantly changing as it is warming up and there is no external heat flux outside of the heatfrom the rising air.

Since data collection took over twelve hours, the system did not have the opportunity to cool completelybetween runs. To compensate, on cool days, the data collection only began after the system warmed up.Even with this precaution, though, the starting temperatures of the individual thermocouples were never thesame. However, the resulting CHTC values differed by a maximum of 4.7 % between comparative runs.

Each run produced eight data curves — one for each thermocouple in one of the two clusters. Simultane-ous measurement of the two data clusters was not possible because of technical limitations of the setup.

For the temperature measurements there are three sources of error: the thermocouples themselves, theamplifier, and the DAQ. The standard K-type thermocouple has an accuracy of ±1.1°C, and the infrared hasan accuracy of 2 % or a maximum of ±1.2°C. According to the literature, the amplifier used has an errorof ±1°C, while the DAQ has a maximum error of ±3.8764 mV which converts to ±0.38°C. Therefore themaximum total error, using the root-sum-square methodology, is ±1.53°C for the K-type thermocouple and±1.61°C for the infrared thermocouple.

Figure 4 shows the temperature data from the bottom cluster when the plenum was 90°C and Fig. 5 showsthe temperature data from the top cluster at the same plenum set point. The first thing that immediatelybecomes apparent is the difference between both the individual temperatures and their trend. With thebottom cluster, the air temperature reached a fairly steady value after approximately three hours whereas,


Fig. 4. Temperature distribution of bottom cluster.

Fig. 5. Temperature distribution of top cluster.

the top cluster never stabilizes but rather keeps increasing throughout the entire 7 hour time period. Thiswas most likely because the top cluster was only 305 mm from the exit so there was a competing downdraftfrom the room which would cause turbulence. Another interesting difference between the top and bottomclusters is the surface temperature. In the bottom cluster the surface temperature is less than 1°C hotter thanthe next thermocouple reading whereas with the top cluster the surface temperature is more than 4°C awayand only just over 2°C from the air temperature. This happened with all the top cluster tests and, given thefact that the surface temperature readings started at roughly the same point, it cannot be a systematic sensorerror. This disparity, as seen below, caused a significant increase in the CHTC as it raises flux values and,more importantly, decreases the temperature difference on which the CHTC is based.


The radiative heat transfer to the room might be a partial reason for the discrepancy between the surfacetemperatures of the top and bottom clusters. To test the plausibility of this, the system was run with theflues closed from external influences. While sealing the flues changes the fluid dynamics, the heat transfermechanism is essentially the same. The results were definitive. With the flue sealed, the surface temperaturewas less than a degree from the next thermocouple in the wall.

4. CALCULATION OF COEFFICIENTS

The equation for convective heat transfer is:

h =q′′

Tw−T∞

. (1)

The raw data provide T∞ and Tw which leaves q′′, the heat flux. There are two negative heat fluxes —conduction through the wall and radiation from the interior wall of the flue. Assuming that the thermalconductivities are uniform throughout their thickness, the transverse one-dimensional Fourier equation is:

q′′con = ke f fT1−T2

∆x. (2)

In this equation, ke f f is the effective thermal conductivity of the section of the wall, T is the temperatureat a specified point, and ∆x is the distance between the two points. To palliate the possible random error fromusing only two points, the best alternative was to calculate multiple fluxes. Effective thermal conductivitieswere calculated for eleven sets of data points — six using the surface as a reference and five using the marbleas a reference — see Table 1 for classification.

Surface-referenced heat fluxes Marble-referenced heat fluxes

Surface to ⅓ terracotta ⅓ terracotta to mid-marble Surface to ⅔ terracotta ⅔ terracotta to mid-marble

Surface to terracotta/mortar interface Terracotta/mortar interface to mid-marble Surface to mid-mortar Mid-mortar to mid-marble

Surface to mortar/marble interface Mortar/marble interface to mid-marble Surface to mid-marble

Table 1

Plenum Bottom cluster CHTC Top cluster CHTC Temperature (°C) (W/m2 ∙#°C) (W/m2 ∙#°C)

60 7.2 ±#10.0 63.6 ±#138 70 7.0 ±#8.9 64.6 ±#104 80 7.6 ±#8.6 69.8 ±#140 90 6.8 ±#9.2 63.2 ±#131

100 6.2 ±#8.4 60.2 ±#86.2

Table 2

Plenum Rayleigh Number Nusselt Number Temperature (°C)

60 −1.65 ×#108 ±#3.1#×#107 154 ±#216 70 −1.49 ×#108 ±#3.0#×#107 146 ±#190 80 −1.64 ×#108 ±#2.8#×#107 162 ±#182 90 −1.45 ×#108 ±#2.7#×#107 142 ±#193

100 −1.65 ×#108 ±#2.5#×#107 130 ±#175

Table 3

Table 1. Breakdown of heat fluxes.

The conduction heat fluxes were calculated for each time increment. Figure 6 shows the resulting con-ductive heat fluxes for the bottom cluster using the surface as the reference in the same sample plenumtemperature above and Fig. 8, using marble as the reference. Figures 7 and 9 show the fluxes for the topcluster. First thing to note for both cases is how quickly the conduction steadies compared to the tempera-ture. By 500 minutes, the flux in both clusters had stopped fluctuating appreciably. This means that, whilethe temperature was still climbing, the amount of heat going into the wall remained constant.

With the uncertainty in the temperature being high relative to the temperature differences in the wall, theuncertainty in the heat flux is large, despite the fact that two of the three variables are actually constants.For both clusters, therefore, the scattering of the heat fluxes can be attributed to experimental error. Thisis most readily apparent in Figs. 6 and 8 where the 1/3 TC curve is separated from the remaining curves.The distance between this thermocouple and the wall surface is just over 4 mm and is the smallest of any ofthe surface-referenced calculations. This means that the heat flux calculation magnifies any error in the 1/3


TC thermocouple reading and gives the appearance of an alternate heat flux value. In addition, there is nodiscernable pattern to the spread, however, the closer together the thermocouples are, the more likely it is forthe resulting fluxes to be more askew because the random error grows relative to the temperature difference.

Fig. 6. Heat fluxes from bottom cluster with the surface temperature as the reference point.

Fig. 7. Heat fluxes from top cluster with the surface temperature as the reference point.

The radiative heat transfer was not easy to estimate. In the Roman bath heating system, each flue wouldbe covered by a horizontal pipe which would take the exhaust gas to a chimney and vent it outside whereasin the experimental setup, the flue was open to the room. This presents two possibilities for the radiationfrom the inner wall. In the bath, the radiation contained within the flue would not transfer much heat asthe walls are within 2°C of each other and there would be little radiation. In the experiment, however, theradiation could diffuse throughout the room thereby cooling the surface.


Fig. 8. Heat fluxes from bottom cluster with the mid-marble temperature as the reference point.

Fig. 9. Heat fluxes from top cluster with the mid-marble temperature as the reference point.


For the purposes of this setup, the heat transfer due to radiation was initially ignored. Therefore, theCHTC was obtained by dividing the heat flux by the difference between the surface temperature and thebulk temperature, refer to Eq. (1). An average was taken as well, excluding those CHTCs that were obvi-ously skewed, namely the one calculated from the heat flux between the mortar/marble interface and themid-marble thermocouples. The results were then graphed versus time and are shown in Figs. 10–13.Figure 10 shows the results from the bottom cluster for a plenum temperature of 90°C using the surface-referenced heat fluxes and Fig. 12 the marble-referenced heat fluxes. Figures 11 and 13 display the CHTCbreakdown for the top cluster. The patterns were nearly identical as those for the fluxes above, particularlywhen the results steadied. The fluctuations in the CHTC came from the fact that the bulk temperature wasnever constant.

Fig. 10. CHTCs using heat fluxes from bottom cluster with the surface temperature as the reference point.

Fig. 11. CHTCs using heat fluxes from top cluster with the surface temperature as the reference point.


Fig. 12. CHTCs using heat fluxes from bottom cluster with the mid-marble temperature as the reference point.

Fig. 13. CHTCs using heat fluxes from top cluster with the mid-marble temperature as the reference point.

There is a considerable difference in the CHTCs from the bottom and top clusters — nearly a factor often. The average of the averages for various plenum temperatures are shown in Table 2.

The uncertainty determined for the CHTC (and Nu latter) appears to be significant. While theoreticallycorrect, however, we feel this does not accurately portray the experimental results. The uncertainty ariseslargely from the fact that the small temperature differentials in the heat flux calculations are of the order ofthe thermocouple errors, as mentioned earlier. When we compared the results of multiple runs for the sameconditions, the standard deviation of the temperature data was less than 0.24°C. Given these runs, done ondifferent days, have this small a standard deviation suggests good experimental repeatability and that thecalculated errors, while possible, were not realized.






Table 1


60 7.2 ±#10.0 63.6 ±#138 70 7.0 ±#8.9 64.6 ±#104 80 7.6 ±#8.6 69.8 ±#140 90 6.8 ±#9.2 63.2 ±#131

100 6.2 ±#8.4 60.2 ±#86.2

Table 2


60 −1.65 ×#108 ±#3.1#×#107 154 ±#216 70 −1.49 ×#108 ±#3.0#×#107 146 ±#190 80 −1.64 ×#108 ±#2.8#×#107 162 ±#182 90 −1.45 ×#108 ±#2.7#×#107 142 ±#193

100 −1.65 ×#108 ±#2.5#×#107 130 ±#175

Table 3

Table 2. Average CHTC for both clusters.

The CHTCs for the bottom cluster are close to other values for natural CHTCs for air, however, the valuesfrom the top are considerably higher. When compared with the sealed flue mentioned above for whichthe average CHTC was 7.7 W/m2·◦C for a trial when the plenum temperature was 90°C, the data from thetop cluster become suspect. Also, for the case under investigation, it is a safe assumption that most of theterracotta tubes in the walls of the bath would be exposed to scenarios much like the lower cluster or thesealed top cluster and not to the open top cluster. For this reason, the data from the bottom cluster will onlybe considered as accurate within the narrow confines of the initial question.

Furthermore, generally, it should be noted that the CHTCs do not vary as a function of plenum tem-perature. This is somewhat surprising as the expectation is for the CHTC to increase somewhat as the airtemperature rises. In fact, if anything, the CHTC peaks when the plenum temperature is 80°C.

5. DISCUSSION

In order to compare these results to the studies mentioned in the introduction, the Nusselt number iscomputed as:

NuL =hLk

. (3)

In this equation, h is the CHTC, L is the characteristic length, and k is the thermal conductivity of the airat, in this case, the bulk temperature. For this particular setup, the best choice for the characteristic lengthis the height of one flue or 607 mm. This is for a number of reasons. One, this arrangement is most similarto channel flow which uses height as the length scale. Two, since there are two clusters of thermocouples,each centred on a flue, it is the natural divisor in the system. Three, that distance allows the Nusselt numberto be typical of the flue. The other key comparator is the Rayleigh number:

Ra =gβ (Tw−T∞)L3

υα. (4)

In this equation, g is the gravitational acceleration, β is the thermal expansion coefficient for air, Tw is thesurface temperature, T∞ is the bulk temperature of the air, L is the characteristic length, υ is the kinematicviscosity of the air, and α is the thermal diffusivity of the air. All air properties were taken at the bulktemperature. Since, however, for this case the wall temperature is lower than the bulk temperature so theRayleigh number is negative. Another notable difference is the temperature at which the thermal propertiesare calculated. Unlike most studies, the driving force of this study is the air so the film temperature is notapplicable. Therefore, the bulk temperature was used to calculate the properties. The Rayleigh and Nusseltnumbers for the bottom cluster are given in Table 3.






Table 1


60 7.2 ±#10.0 63.6 ±#138 70 7.0 ±#8.9 64.6 ±#104 80 7.6 ±#8.6 69.8 ±#140 90 6.8 ±#9.2 63.2 ±#131

100 6.2 ±#8.4 60.2 ±#86.2

Table 2


60 −1.65 ×#108 ±#3.1#×#107 154 ±#216 70 −1.49 ×#108 ±#3.0#×#107 146 ±#190 80 −1.64 ×#108 ±#2.8#×#107 162 ±#182 90 −1.45 ×#108 ±#2.7#×#107 142 ±#193

100 −1.65 ×#108 ±#2.5#×#107 130 ±#175

Table 3

Table 3. Rayleigh and Nusselt numbers for the bottom cluster based on the average CHTCs.

The first thing that becomes clear is that there is no relationship between the Rayleigh and Nusselt num-bers. This is most likely because the Rayleigh, or more specifically the Grashof, number is a comparativemeasure between buoyancy and viscous forces and, since the air is actually cooled by the walls, it does notcapture the mechanics of this setup even though it is a natural convection system.

If one just looks at the Nusselt numbers, however, there are a few things to note. The pattern of peakingat 80°C present in the CHTCs is more exaggerated here. These Nusselt numbers are more than double thanall but one of the cases mentioned above. Interestingly, the one case (the channel flow studied by Hatamiand Bahadorinejad) that is higher is the closest comparison. When compared to these, the Nusselt numbershere are about half. This setup, which is approximately two millennia old in design, is, therefore, half waybetween the comparable modern equivalents which may be surprising. In essence, the system of terracottapipes is not only incredibly effective for its age, it matches modern counterparts.

6. CONCLUSION

The goal of this project was to determine the convective heat transfer coefficient (CHTC) for terracottaflues designed to replicate the heating system inside an ancient Roman bath. The unique aspect aboutthese experiments was that the air was heated before being exposed to the faux wall and then allowed torise through the flues, thereby warming the flue walls. The five plenum temperatures were tested (60°C,70°C, 80°C, 90°C, 100°C) and we found that the CHTC was relatively insensitive to plenum temperature.The average CHTC was 7.0 W/m2·◦C. Furthermore, there was no relationship established connecting theNusselt number which ranged from 1.3 × 102 to 1.6 × 102 with the Rayleigh number which ranged from−1.7 × 108 to −1.5 × 108.

ACKNOWLEDGEMENTS

The authors acknowledge Dr. David Wood for his help in the preparation of this manuscript as well as thesupport of the Alberta Ingenuity Fund Graduate Student Scholarship program.

REFERENCES

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